* Binomial expression* an expression (a + b)

^{n}which is the sum of two terms raised to the power n.

e.g. (x + 3)^{2}

* Binomial expansion* (a + b)

^{n}expanded into a sum of terms

e.g. x^{2 }+ 6x + 9

Binomial expansions get increasingly complex as the power increases:

The general formula for each term in the expansion is ^{n}C_{r} a^{n-r} b^{r} .

In order to find the full binomial expansion of a binomial, you have to determine the coefficient ^{n}C_{r} and the powers for each term. The powers for an and b are found as n − r and r respectively, as shown by the binomial expansion formula.

#### Binomial expansion formula

**DB 1.9**

The powers decrease by 1 for a and increase by 1 for b for each subsequent term.

The sum of the powers of each term will always = n.

There are two ways to find the coefficients: with Pascal’s triangle or the binomial coefficient function (nCr). You are expected to know both methods.

#### Pascal’s triangle

Pascal’s triangle is an easy way to find all the coefficients for your binomial expansion. It is particularly useful in cases where:

- the power is not too high (because you have to write it out manually)
- you need to find all the terms in a binomial expansion

#### Binomial coefficient functions

The alternative is to calculate the individual coefficients using the nCr function on your calculator, or with the formula below.

(Note:In the 1^{st} term of the expansion r = 0, in the 2^{nd} term r = 1, . . .)

Expanding binomial expressions

Use the binomial expansion formula

2. Find coefficients using Pascal’s triangle for low powers or nCr on calculator for high powers

3. Put the terms and their coefficients together

4. Simplify using laws of exponents

Finding a specific term in a binomial expansion

Find the coefficient of x^{5} in the expansion of (2x − 5)^{8}

Use the binomial expansion formula

^{n}=··· +

^{n}C

_{r }a

^{n−r }b

^{r}+……

2. Determine r

Since a = 2x, to find x^{5} we need a^{5}.

a^{5} = a^{n−r} = a^{8−r}, so r = 3

3. Plug r into the general formula

^{n}C

_{r }a

^{n−r }b

^{r}=

^{8}C

_{3 }a

^{8−3 }b

^{3 }=

^{8}C

_{3 }a

^{5 }b

^{3}

4. Replace a and b

^{8}C_{3 }(2x)^{5 }(−5)^{3}

5. Use nCr to calculate the value of the coefficient, ^{n}C_{r}

^{8}C

_{3}= 8C3 = 56

6. Substitute and simplify

56 × 2^{5}(x^{5}) × (−5)^{3} = −224000(x^{5})

⇒ coefficient of x^{5} is −224000

The IB use three different terms for these types of question which will effect the answer you should give.

* Coefficient* the number before the x value

*the number and the x value*

**Term***the number for which there is no x value (x*

**Constant term**^{0})